Current design programs use two main methods to achieve the desired aerodynamic characteristics for the airfoil. In the direct method, an iterative procedure is used to successively modify the shape of an airfoil until the desired aerodynamic characteristics are achieved. In the inverse method, the desired aerodynamic characteristics are specified and the airfoil shape that meets these objectives is determined.
The set of governing equations used to solve the problem is generally a simplified version of the full Navier-Stokes equations (see Kuethe, A. M. and Chow, C. Y.: Foundations of Aerodynamics, John Wiley & Sons (1998)), which require large computer power and computing time even for simple cases. The Euler equation is obtained when the fluid viscosity is neglected, and the Laplace equation results when the flow is assumed to be irrotational. In the interest of simplicity and in order to reduce the demands on computational resources and time, most airfoil design and analysis methods use either the Euler or Laplace equations. The velocity and pressure fields are determined by solving these equations. The aerodynamic forces and moments are then related to the pressure distribution around the airfoil.
In order to solve the problem for given free-stream conditions, it is necessary to prescribe the boundary conditions at the fluid-structure interface on the airfoil surface. They are usually specified either in terms of the flow direction tangential to the airfoil surface (the “Dirichlet condition”) or in terms of the normal derivatives of the flow at the airfoil surface (the “Neumann condition”). See Woods, L. C.: The Theory of Subsonic Plane Flow, Cambridge University Press, (1961) (“Woods Reference”). A unique solution is usually obtained by imposing the condition that (1) the flow leaves the trailing edge smoothly and (2) the velocity is zero or finite at the trailing edge. See Katz, J. and Plotkin, A.: Low-Speed Aerodynamics, Cambridge University Press, (2001). This condition is generally known as the “Kutta condition.” The viscous effects using various boundary layer approximations and modeling of phenomena such as laminar-to-turbulent transition are then superposed on the potential flow solution in an iterative manner to determine the resultant velocity and pressure field around the airfoil. The aerodynamic forces and moments generated by the airfoil are then obtained by integrating the pressure around the airfoil.
The procedure is repeated to obtain the aerodynamic characteristics at other flow conditions.
A number of computer programs and associated software have been developed using such methods. The differences between various programs may be generally attributed to the relative quality of the viscous modeling used to determine boundary layer characteristics. These design methods have been used to design airfoils with predictable aerodynamic characteristics within a certain range of Reynolds numbers at low angles of attack. However, errors in the computed performance are known to increase significantly in the presence of flow separation near stall and post-stall conditions. Further, the generally used methods fail to predict hysteresis when an airfoil undergoes pitching oscillations.
The most widely used methods for airfoil design and analysis start with a potential flow model and incorporate viscous effects in one of two ways: (1) Eppler's PROFILE (see Eppler, R.: Airfoil Design and Data, Springer-Verlag (Berlin), 1990); and (2) Drela's XFOIL airfoil analysis method (see Drela, M., and Giles, M. B.: ISES: A Two-Dimensional Viscous Aerodynamic Design and Analysis Code, AIAA Paper 87-0424, January 1987). In the original Eppler airfoil analysis method, the pressure distribution is first obtained with a potential flow model. This pressure distribution is used to compute viscous effects using boundary layer approximations and superimposed on the potential flow solution to account for changes in pressure distribution due to viscous effects. The aerodynamic coefficients are then computed from the resulting pressure distribution. An alternate approach used in Drela's XFOIL code attempts to interactively couple viscous effects with the potential flow solution. Most other methods follow similar basic concepts.
The potential flow method is based on solving the governing Laplace equation, which is a second-order linear partial differential equation. The flow direction (α) in the free stream in relation to a reference axis is assumed known. It requires two independent boundary conditions to obtain, out of infinite possible combinations, one unique solution.
Two types of boundary conditions are generally considered. In the Neumann type conditions, zero-flow normal to a solid surface is specified. In the Dirichlet type conditions, the flow about a body is uniquely determined if the flow direction θ is completely known on the given profile surface. The flow direction θ is measured from the chord line direction, a convenient reference line fixed in airfoil. If θS is the value of θ on the airfoil surface, then if the leading edge stagnation point (LESP) S1 and the flow separation point (FSP) S2 are fixed in position on the airfoil surface, θS is completely known, and the flow direction at all points in the flow is fixed. In particular, θ∞, the flow direction at ∞, is fixed, as the incidence α is the angle between the airfoil chord (reference line) and the stream direction at ∞, α=θ∞. Thus corresponding to each pair of locations S1, S2 there is a unique value of α. The pressures over the airfoil are also uniquely determined, and therefore the lift L is fixed in value. Conversely, if L and alpha are prescribed the positions S1 and S2 will be uniquely determined. More generally we need to prescribe any pair of the quantities alpha, L, S1, S2 to obtain a unique Dirichlet flow (see Woods Reference).
In both cases, the problem is usually resolved by assuming Kutta condition at the trailing edge. The Kutta condition at the trailing edge simultaneously provides the basis for calculating the circulation generated by the lifting body. Such an approach has been known and used for over a century and has provided an excellent first approximation for the aerodynamic forces and moments generated by a lifting body (see Theodorsen, T. and Garrick, I. E.: General Potential Theory of Arbitrary Wing Sections. NACA TR 452 (1933)), when the lift coefficient varied almost linearly with α. This approach gave poor results in the presence of extensive flow separation on the suction side of the airfoil upstream of the trailing edge at high α, near and beyond stall (see de Vargas, L. A. T., de Oliveira, P. H. I., de Freitas Pinto, R. L. U., Bortoulus, M. V., and e Souza, M. da Silva.: Comparison between modern procedures for aerodynamic calculation of subsonic airfoils for applications in light aircraft designs, Proc. of COBEM, 18th International Congress of Mechanical Engineering, November 2005, Ouro Preto, MG.).
The current methods suffer from a fundamental mathematical drawback: the leading-edge stagnation point obtained initially by imposing the Kutta condition at the trailing edge is not fully corrected in response to viscous corrections that are computed in an iterative manner. Since the potential flow model is governed by elliptic equations (see Jeffrey, A.: Applied Partial Differential Equations. An Introduction. Elsevier Science, (2002)), where the downstream conditions have an impact on the upstream flow conditions, the presence of flow separation has a direct impact on the location of the leading-edge stagnation point and vice versa. This in turn has a significant impact on the aerodynamic forces and moments generated by the airfoil. In the absence of extensive flow separation (e.g., at low angles of attack), the error introduced by assuming a fixed leading-edge stagnation point location is marginal and practically ignored both in experiments and computations. However, the error becomes significant at high angles of attack, especially near and beyond stall. As a result, these methods completely fail at and beyond stall when there is a real potential for generating high lift.
Thus the conventional potential flow model provides useful but restricted solutions due to viscous effects that manifest themselves in the boundary layer in the form of transition, turbulence, and flow separation. On the other hand, full Navier-Stokes (N-S) equations could be used to solve the problem numerically but they require tremendous computational resources and time. Even then, a degree of empiricism is required to model and predict transition, turbulence, and flow separation in solving N-S equations. Thus, strictly speaking, the N-S approach is semi-empirical in nature, though significantly more detailed and complex compared to the potential flow approach. In comparison, the potential flow approach is simpler and, with proper physics and accurate mathematical modeling, it is capable of providing valuable engineering solutions with minimal computing resources and time.